📚 Tackling p288_1.pdf: Angle Problems in Triangles | 解析 p288_1.pdf：三角形角度问题
e unknown angle in a triangle when two angles are given as algebraic expressions and one angle is a constant. Understanding how to set up and solve the resulting equation is an essential skill for the KS3 Cambridge Mathematics course.
本文讲解 p288_1.pdf 中一道典型的剑桥 KS3 数学题。题目要求在已知两个角为代数式、一个角为常数的三角形中求出未知角。掌握如何建立并求解由此得出的方程是 KS3 剑桥数学课程中的一项重要技能。
1. Understanding Angle Sum in a Triangle | 理解三角形内角和
In any triangle, the three interior angles always add up to 180 degrees. This is a key fact used in many KS3 geometry questions.
在任何三角形中，三个内角的和始终为 180 度。这是许多 KS3 几何题中使用的关键事实。
Angle A + Angle B + Angle C = 180°
No matter whether the triangle is scalene, isosceles or equilateral, this rule always holds. It is often called the angle sum of a triangle.
无论三角形是不等边三角形、等腰三角形还是等边三角形，这个规律始终成立。它通常被称为三角形的内角和定理。
- Equilateral triangle: all angles are 60°, so 60 + 60 + 60 = 180.
- Right-angled triangle: one angle is 90°, so the other two must add up to 90°.
- Scalene triangle: three different angles, but they still add up to 180°.
- 等边三角形：所有角都是 60°，所以 60 + 60 + 60 = 180。
- 直角三角形：一个角是 90°，因此另外两个角之和必须为 90°。
- 不等边三角形：三个角各不相同，但它们的和仍然是 180°。
2. Reading the Given Information | 阅读题目信息
In p288_1.pdf, the problem states: "The three angles of a triangle are x°, 2x° and 60°. Find x." Before doing any calculation, we need to identify the three angle expressions clearly.
在 p288_1.pdf 中，题目为：“三角形的三个角分别为 x°、2x° 和 60°，求 x。”在进行任何计算之前，我们需要清楚地识别这三个角的表达式。
| Angle | Expression in degrees |
| First angle | x° |
| Second angle | 2x° |
| Third angle | 60° |
The letter x is an unknown number. Once we find x, we can find the value of every angle in the triangle.
字母 x 是一个未知数。一旦我们求出 x，就能求出三角形中每个角的度数。
3. Setting Up the Equation | 建立方程
Because the three angles add up to 180 degrees, we can write an equation. Replace the words "angle sum" with the actual expressions.
因为三个角之和为 180 度，我们可以写出一个方程。用实际的表达式代替“内角和”这样的文字。
x + 2x + 60 = 180
This equation is linear because the highest power of x is 1. Linear equations can be solved by balancing both sides.
这是一个线性方程，因为 x 的最高次数是 1。线性方程可以通过平衡两边来求解。
- The left-hand side has three terms: x, 2x and 60.
- The right-hand side is the total angle sum, 180.
- Our goal is to isolate x on one side.
- 左边有三项：x、2x 和 60。
- 右边是总的内角和 180。
- 我们的目标是将 x 单独放在一边。
4. Combining Like Terms | 合并同类项
The terms x and 2x are called like terms because they both have the same variable x. We can add them together before solving.
项 x 和 2x 被称为同类项，因为它们都含有相同的变量 x。我们可以在求解之前先将它们相加。
x + 2x = 3x
So the equation can be rewritten in a simpler form. This makes the next step much easier.
因此，这个方程可以改写为更简单的形式。这样下一步会容易得多。
3x + 60 = 180
Always check that you have combined only like terms. The constant 60 cannot be added to 3x because it has no x.
务必检查你是否只合并了同类项。常数 60 不能与 3x 相加，因为它不含 x。
5. Solving for x | 求解 x
e equation 3x + 60 = 180. First subtract 60 from both sides to keep the equation balanced.
现在我们解方程 3x + 60 = 180。首先从两边同时减去 60，以保持方程平衡。
3x + 60 - 60 = 180 - 60
This simplifies to the following equation. The 60 on the left cancels out, leaving only the x-term.
化简后得到以下方程。左边的 60 相互抵消，只剩下含 x 的项。
3x = 120
Finally, divide both sides by 3 to find x. This gives the value of the unknown.
最后，两边同时除以 3，求出 x。这样就得出了未知数的值。
x = 120 ÷ 3 = 40
So x is 40. We must remember that x is measured in degrees here because it represents an angle.
所以 x 等于 40。我们必须记住，这里 x 的单位是度，因为它代表一个角。
6. Finding Each Angle | 求出每个角
te this value back into the original expressions. This gives the size of each angle in the triangle.
现在我们知道 x = 40，可以将这个值代回原来的表达式。这样就得到了三角形中每个角的大小。
| Angle expression | Substitution | Value |
| x | 40 | 40° |
| 2x | 2 × 40 | 80° |
| constant | 60 | 60° |
The three angles are therefore 40°, 80° and 60°. The triangle is scalene because all three angles are different.
因此，这三个角分别为 40°、80° 和 60°。这个三角形是不等边三角形，因为三个角都不相同。
7. Checking Your Answer | 检查答案
A final check is always worth doing. Add the three angles together to make sure they equal 180 degrees.
最后检查一下总是值得的。将三个角相加，确保它们等于 180 度。
40° + 80° + 60° = 180°
The total is exactly 180 degrees, so our solution is correct. If the sum were not 180, we would need to go back and check our working.
总和正好是 180 度，所以我们的解答是正确的。如果和不等于 180，我们就需要返回去检查计算过程。
Checking also helps us avoid simple arithmetic errors. In an exam, this habit can earn you valuable marks for showing clear reasoning.
检查还能帮助我们避免简单的计算错误。在考试中，这个习惯可以让你因为展示清晰的推导过程而获得宝贵的分数。
8. Common Mistakes to Avoid | 常见错误避免
There are several common errors students make when solving angle equations in triangles. Being aware of them will help you avoid losing marks.
学生在求解三角形角度方程时有几个常见错误。了解这些错误有助于你避免丢分。
- Forgetting the angle sum: Using 90° or 360° instead of 180° for a triangle.
- Incorrect combining: Trying to add x and the number 60 as if they were like terms.
- Missing a term: Leaving out one of the three angles when writing the equation.
- Not solving for all angles: Finding x but forgetting to multiply by 2 to find 2x.
- 忽略内角和： 对三角形使用 90° 或 360° 而不是 180°。
- 错误合并： 试图将 x 和数字 60 当作同类项相加。
- 遗漏某一项： 写方程时漏掉三个角中的一个。
- 没有求出所有角： 求出了 x，却忘记乘以 2 来求 2x。
Always re-read the question after solving to make sure you have answered the exact question asked.
解题后务必再读一遍题目，确保你回答的正是题目所问的内容。
9. Practice Questions | 练习题目
Try these similar questions to build confidence. The same method can be used for each one.
尝试以下类似的题目来建立信心。每一道题都可以使用相同的方法。
- 1. The angles of a triangle are x°, 3x° and 90°. Find x and state all three angles.
- 2. In a triangle, the angles are 2x°, 4x° and 60°. Find x and check your answer.
- 3. A triangle has angles x°, x + 20° and 100°. Find x.
- 1. 三角形三个角分别为 x°、3x° 和 90°。求 x 并写出三个角。
- 2. 在一个三角形中，三个角分别为 2x°、4x° 和 60°。求 x 并检查答案。
- 3. 一个三角形的三个角分别为 x°、x + 20° 和 100°。求 x。
uation is x + 3x + 90 = 180, so 4x = 90 and x = 22.5. The angles are 22.5°, 67.5° and 90°.
第 1 题的方程是 x + 3x + 90 = 180，所以 4x = 90，x = 22.5。三个角分别为 22.5°、67.5° 和 90°。
e equation is 2x + 4x + 60 = 180, so 6x = 120 and x = 20. The angles are 40°, 80° and 60°.
第 2 题的方程是 2x + 4x + 60 = 180，所以 6x = 120，x = 20。三个角分别为 40°、80° 和 60°。
100 = 180, so 2x + 120 = 180, giving 2x = 60 and x = 30. The angles are 30°, 50° and 100°.
第 3 题的方程是 x + x + 20 + 100 = 180，所以 2x + 120 = 180，得到 2x = 60，x = 30。三个角分别为 30°、50° 和 100°。
10. Exam Tips | 考试技巧
When you see a triangle with angles expressed in terms of x, always start by writing the angle sum equation. Show every step clearly so the examiner can follow your method.
当你看到三角形的角用 x 表示时，始终先写出内角和方程。清楚地展示每一步，以便阅卷老师能够看懂你的解题方法。
Use the formula below as a quick reminder. It is simple but extremely useful.
使用下面的公式作为快速提醒。它很简单，但非常有用。
Sum of angles in a triangle = 180°
If the question involves a quadrilateral, remember that the angle sum is 360°. However, for most KS3 Cambridge triangle questions, 180° is the key number.
如果题目涉及四边形，请记住内角和为 360°。不过，对于大多数 KS3 剑桥三角形题目来说，180° 才是关键数字。
Finally, always check your substitution at the end. A quick addition of the three angles takes only a few seconds and confirms that your answer makes sense.
最后，在结束时一定要检查你的代入结果。快速把三个角加起来只需几秒钟，就能确认你的答案是否合理。
Published by TutorHao | KS3 Cambridge Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图